##### Find the x-value(s) of the relative maxima and relative minima

label Calculus
account_circle Unassigned
schedule 1 Day
account_balance_wallet \$5

Jul 15th, 2015

Thank you for the opportunity to help you with your question!

First, we rewrite the 32/x by using this power law:   1/x = 1/x^1 = x^-1

f(x) = 2x^2 + 32/x + 7  ------------> f(x) = 2x^2 + 32x^-1 + 7

Then we find the derivative of the function. Let's remember the derivative of a power:   (x^n)' = n*x^n-1 ; where n is the exponent of the x. By the way, let's remember also that the derivative of any constant is 0.

f'(x) = (2x^2 + 32x^-1 + 7)'  ------> f'(x) = (2x^2)' + (32x^-1)' + (7)' ----> f'(x) = 2(x^2)' + 32(x^-1)' + (7)'

f'(x) = 2(2x^2-1) + 32(-1x^-1-1) + 0  --------> f'(x) = 4x^1 - 32x^-2   --------> f'(x) = 4x - 32x^-2

f'(x)= 4x - 32/x^2

Then we equal the derivative to zero and solve for x.

4x - 32/x^2 = 0  -----------> 4x*x^2 - 32/x^2 * x^2 = 0 ----------->  4x^3 - 32 = 0 ------------> 4x^3 - 32 + 32 = 0 + 32

4x^3 = 32  ----------->  4x^3/4 = 32/4 ------------> x^3 = 8   ------------>  (x^3)^1/3 = (8)^1/3  --------> x = 8^1/3

Or x = cubic root(8) ---------->  x = 2

Then we find the second derivative.

f''(x) = (4x - 32x^-2)' = (4x)' - (32x^-2)' = 4(x)' - 32(x^-2)' = 4(1) - 32(-2)x^-2-1 = 4 + 64x^-3

f''(x) = 4 + 64/x^3

Then we enter x = 2 into the second derivative and if it is greater than zero (positive) then it is a minimum

and if it is less than zero (negative) then it is a maximum.

f''(2) = 4 + 64/2^3 = 4 + 64/8 = 4 + 8 = 12 > 0 (positive). Then x = 2 is a minimum and we don't have any maximum.

Relative maxima: DNE (we don't have it).

Relative minima: x = 2

Please let me know if you have any doubt or question.

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jul 15th, 2015

...
Jul 15th, 2015
...
Jul 15th, 2015
Sep 26th, 2017
check_circle
Mark as Final Answer
check_circle
Unmark as Final Answer
check_circle
Final Answer

Secure Information

Content will be erased after question is completed.

check_circle
Final Answer